Fermion Doubling in Quantum Cellular Automata

Abstract

A Quantum Cellular Automaton (QCA) is essentially an operator driving the evolution of particles on a lattice, through local unitaries. Because $Δ_t=Δ_x = ε$, QCAs constitute a privileged framework to cast the digital quantum simulation of relativistic quantum particles and their interactions with gauge fields, e.g., $(3+1)$D Quantum Electrodynamics (QED). But before they can be adopted, simulation schemes for high-energy physics need prove themselves against specific numerical issues, of which the most infamous is Fermion Doubling (FD). FD is well understood in particular in the real-time, discrete-space \emph{but} continuous-time settings of Hamiltonian Lattice Gauge Theories (LGTs), as the appearance of spurious solutions for all $Δ_x=ε\neq 0$. We rigorously extend this analysis to the real-time, discrete-space \emph{and} discrete-time schemes that QCAs are. We demonstrate the existence of FD issues in QCAs for $Δ_t =Δ_x = ε\neq 0$. By applying a covering map on the Brillouin zone, we provide a flavor-staggering-only way of fixing FD that does not break the chiral symmetry of the massless scheme. We explain how this method coexists with the Nielsen-Ninomiya no-go theorem, and give an example of neutrino-like QCA showing that our model allows to put chiral fermions interacting via the weak interaction on a spacetime lattice, without running into any FD problem.